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Normed Vector Space

Module by: Phil Schniter

Summary: This module introduces normed vector space.

Now we equip a vector space VV with a notion of "size".

  • A norm is a function ( ∥⋅∥:V→ℝ : ⋅ V ) such that the following properties hold ( ∀x,y,x∈V∧y∈V x y x V y V and ∀α,α∈ α α ):
    1. ∥x∥≥0 x 0 with equality iff x=0 x 0
    2. ∥αx∥=|α|⋅∥x∥ α x α ⋅ x
    3. ∥x+y∥≤∥x∥+∥y∥ x y x y , (the triangle inequality).
    In simple terms, the norm measures the size of a vector. Adding the norm operation to a vector space yields a normed vector space. Important example include:
    1. V=ℝN V N , ∥ x 0 … x N - 1 T∥≔∑i=0N-1 x i 2≔xTx ≔ x 0 … x N - 1 i 0 N 1 x i 2 x x
    2. V=ℂN V N , ∥ x 0 … x N - 1 T∥≔∑i=0N-1| x i |2≔xHx ≔ x 0 … x N - 1 i 0 N 1 x i 2 x x
    3. V= l p V l p , ∥xn∥≔∑n=-∞∞|xn|p1p ≔ x n n x n p 1 p
    4. V= ℒ p V ℒ p , ∥ft∥≔∫-∞∞|ft|pdt1p ≔ f t t f t p 1 p

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